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## Does the Manin-Drinfeld theorem hold over number fields?

The Manin-Drinfeld theorem has various equivalent statements. Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Then:

• for any congruence subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$, the subgroup of $\operatorname{Jac}(X(\Gamma))$ generated by the cusps is finite;
• if $c_1, c_2$ are cusps and $f \in S_2(\Gamma)$ is a Hecke eigenform, the integral $\int_{c_1}^{c_2} f(z) \mathrm{d}z$ is a linear combination of the periods $\Omega_+(f)$ and $\Omega_-(f)$ with coefficients in the field generated by the Hecke eigenvalues of $f$;
• the natural map $H^1_c(Y(\Gamma), \mathbb{Q}) \to H^1(Y(\Gamma), \mathbb{Q})$ has a unique Hecke-equivariant section.

I'm interested in extensions of this to the context of arithmetic quotients of $\mathrm{GL}_2(\mathbb{A}_K)$, where $K$ is a number field. The first statement only makes sense if $K$ is totally real, but the second and third can be formulated for any $K$. Are they true in this generality?

(I have a translation of a 1978 paper by Kurcanov which gives the proof for $K$ an imaginary quadratic field; and I believe there is a more recent paper of Kurcanov that covers CM fields, but I can't read Russian and there doesn't seem to be an English translation.)

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 I believe there is a paper by Harder in the Janos Bolyai conference (early 1070?) which deals with these questions. – Aakumadula Mar 14 at 12:04 Harder's paper contains lots of relevant-looking statements on cohomology, compactly-supported cohomology, and cuspidal cohomology, but he doesn't seem to address this question directly as far as I can see. – David Loeffler Mar 14 at 12:24 the Hecke action on the image of compactly supported cohomology has eigenvalues of absolute value different from those on the eisenstein cohomology and hence there is a natural hecke equivariant section. I believe this is done in that paper of Harder. – Aakumadula Mar 14 at 12:56 I have the paper in front of me, and it contains nothing whatsoever about eigenvalues of Hecke operators. – David Loeffler Mar 14 at 13:50